 StatLect

# Complex vectors and matrices

Up to this point, we have progressed in our study of linear algebra without ever specifying whether the entries of our vectors and matrices are real or complex numbers. Although the examples and exercises presented thus far concern real matrices (i.e., matrices having real entries), all the definitions, propositions and results found in previous lectures are applicable without modification to complex matrices (i.e., matrices whose entries are complex numbers). In fact, if you revise those lectures, you will realize that nowhere, and especially in no proof, it is necessary to assume that a matrix or vector be real. The only caveat is that when we deal with complex matrices, we also need to use complex scalars when taking linear combinations.

In this lecture, we are going to revise some elementary facts about complex numbers. We then show some basic properties of complex matrices and provide some useful definitions. ## Complex numbers

A complex number is a number that can be written as where and are real numbers, called the real and imaginary part of the complex number respectively, and is called imaginary unit.

Hence, when we manipulate complex numbers, the key "trick" we exploit over and over again is Imaginary numbers allow us to find solutions to equations that have no real solutions. For example, the equation has no real solution, but it has two imaginary solutions Real numbers are complex numbers that have zero imaginary part. The latter is often omitted, that is, instead of writing we simply write .

### Complex conjugate

An important concept is that of complex conjugate. Given a complex number its conjugate, denoted by , is As a consequence, double conjugation leaves numbers unchanged: ### Algebra of complex numbers

The algebra of complex numbers is similar to the algebra of real numbers. Given two complex numbers we have the following rules:

1. Addition: 2. Subtraction: 3. Multiplication: 4. Division: ### Distributive properties of conjugation

Note that conjugation is distributive under addition: and under multiplication: ### Modulus of a complex number

The modulus (or absolute value) of a complex number is defined as where we consider only the positive root.

Clearly, the modulus is always a real number.

## Complex matrices

Complex matrices (and vectors) are matrices whose entries are complex numbers.

### Complex conjugation of matrices

Given a matrix , its complex conjugate is the matrix such that that is, the -th entry of is equal to the complex conjugate of the -th entry of , for any and .

Example Define the matrix Then its complex conjugate is ### Distributive properties of conjugation

The distributive properties that hold for the conjugation of complex numbers hold also for the conjugation of matrices.

Proposition If and are two matrices, then Proof

We have that for any and , by the distributive property of the conjugation of complex numbers under addition.

Proposition If is matrix and is a matrix, then Proof

We have that for any and , by the distributive property of the conjugation of complex numbers under addition and multiplication.

Proposition If is a matrix and is a scalar, then Proof

We have for any and , by the definition of multiplication of a matrix by a scalar and by the distributive property of the conjugation of complex numbers under multiplication.

### Conjugate of a real matrix

A trivial but useful property is that taking the conjugate of a matrix that has only real entries does not change the matrix. In other words, if has only real entries, then This is a consequence of the fact that a real number can be seen as a complex number with zero imaginary part. But all that conjugation does is to change the sign of the imaginary part of a complex number. Therefore, a real number is equal to its conjugate.

## Solved exercises

Below you can find some exercises with explained solutions.

### Exercise 1

Define two vectors Compute the following modulus: Solution

The product of and is and its modulus is ### Exercise 2

Define and Compute Solution

First of all, we can substitute into : The complex conjugate of is The product we need to compute is 